Abstract
In this paper, we establish the solution of the fourth-order nonlinear homogeneous neutral functional difference equation. Moreover, we study the new oscillation criteria have been established which generalize some of the existing results of the fourth-order nonlinear homogeneous neutral functional difference equation in the literature. Likewise, a few models are given to represent the significance of the primary outcomes.
1. Introduction
The investigation of the conduct of solutions of functional difference equations is a significant space of examination and is quickly developing because of the advancement of time scales and the time-scale analytics (see, e.g., [1, 2]). Most papers on higher-order nonlinear neutral difference equations manage the presence of positive solutions and the asymptotic conduct of solutions. We refer the per user to a portion of the works [3–9] and the references referred to in that.
In [10], Migda investigated the asymptotic properties of nonoscillatory solutions of neutral difference equation of the form
The papers [9, 10] are tantamount. Be that as it may, more accentuation might be given to [9], which manages the oscillatory, nonoscillatory, and asymptotic characters. Some oscillation criteria have been set up by applying the discrete Taylor series [11]. The inspiration of the current work has come from two bearings [9, 12] and the second is expected to [13].
In 2008, Tripathy [9] investigated oscillatory and asymptotic behaviour of solutions of a class of fourth-order nonlinear neutral difference equations of the form
Also,
Under the condition , for different ranges of and . Also, no super linearity or sublinearity conditions are imposed on . It is interesting to observe that the nature of the function influences the behaviour of solutions of (2) or (3).
In 2013, Tripathy [12] discussed oscillatory behaviour of solutions of a class of fourth-order neutral functional differences (2) and (3) under the conditions and .
Parhi and Tripathy [13] studied oscillation of a class of nonlinear neutral difference equations of higher order and the behaviour of its solutions is studied separately.
In this present work, we study the oscillation behaviour of the fourth-order nonlinear homogeneous neutral functional difference equation.
2. Oscillation Behaviour of Neutral Difference Equation
In this section, we establish the solution of the fourth-order homogeneous neutral functional difference equation of the formwhere(1) is a difference operator(2)(3) are real valued functions defined on (4) is nondecreasing(5) are constants with
A solution of (4) is oscillatory if for every integer . Otherwise, it is nonoscillatory. The NFDE (4) is oscillatory, if all its solutions are oscillatory.
For the oscillation of (4), we define the operators
Also, the notations
Lemma 1. Let (5) hold andbe a real valued function withfor large. If, then one of these (8) to (9) holds and if, then one of these (9) to (13) holds, where
Lemma 2. Letand Lemma 1hold. Then,for, where
Theorem 1. Let. If (5) andhold, then (4) is oscillatory.
Proof. Let be a nonoscillatory solution of (4) and for .
If we setthen (4) becomesHence, we find such that are eventually of one sign on .
Consider the potential cases 2.3 to 2.6 of Lemma 1.
Case 2.5. For , it follows from the Taylor series thatAlso,which impliesThat is,Consequently,For , the above disparity can be composed asUsing (4), it is not difficult to check thatExpected to (16), (17) and (31) giveThat is,As a result,which is logical inconsistency to (21) due to (15).
Case 2.6. For , it follows from (25) thatSincewe haveThat is,Consequently,For , the above disparity can be composed asApplying (16) and (17) in (31) yields thatExpected to (40), the above equation becomesTherefore,which is logical inconsistency to (22) due to (15).
Cases 2.3 and 2.4 can be managed also as the above cases.
Finally, for . Using (16) and in (4), we obtain andProceeding as above, we see that every solution of (44) oscillates.
This completes the proof of theorem.
Theorem 2. Letand. If (5) to (16) andhold, then every solution of (4) oscillates.
Proof. Let the opposition that is a nonoscillatory solution of (4) and for .
The case for can be dealt with similarly.
Setting by (23), we get (24).
Consequently, we find such that and are eventually of one sign on .
Let . Then, for and (24) becomesApplying Lemma 1 to (52) and Theorem 1, we get contradictions to (45) to (48) expected to .
Next, for and for implies .
By Lemma 1, the cases of 2.4 to 2.8 hold for.
Case 2.4. Sinceit follows thatFor , summing the above inequality from to , we haveThat is,which impliesfor .
Therefore, for ,Since (4) can be viewed asusing (58), (59), and (16) yieldsSumming the last inequality form to , we getHence,which gives the contradiction to (49).
Case 2.5. From (25), we havefor andwhich implies thatThat is,Consequently,Also hence,holds for , using (16) and (68) in (59) and summing from to , we obtainTherefore,which is logical inconsistency to (50).
Case 2.6. We use (25) and it follows thatfor . Sincewe havewhich implies thatHence, for , it follows thatConsequently, (59) becomesAs a result,which is logical inconsistency to (51).
In both cases 2.7 and 2.8, we see that .
On the other hand, for implies for .
That is,Hence, is bounded and is bounded, which is a contradiction.
This completes the proof of the theorem.
Example 1. Consider the NFDE,Here,Clearly, all conditions of Theorem 1 are satisfied.
Hence, (79) is oscillatory and is one of the oscillatory solutions of (79).
3. Conclusion
In this paper, we inferred new properties of the nonoscillatory solutions and using these outcomes, some new adequate are introduced for the concentrated on NFDE to have the purported property oscillatory. Our outcomes improve and supplement many known outcomes for NFDEs as well as for ordinary functional difference equations also. At last, we give two models that show the meaning of the fundamental results.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.